An abelianization of the SU(2) WZW model
type: Item
instance of: scholarly article
title: An abelianization of the SU(2) WZW model
zbMATH Open document ID: 1112.14040
author: Tomoyoshi Yoshida
published in: Annals of Mathematics. Second Series
publication date: 2007-01-04T00:00:00Z
review text: Let \(M_g\) be the moduli space of semi-stable holomorphic vector bundles of rank 2 on a Riemann surface \(C\) of genus \(g\). Denote by \(\mathcal{L}\) the determinant line bundle on \(M_g\). An element of the space of holomorphic sections \(H^0(M_g,\mathcal{L}^k)\) is called a a rank \(2\) theta function of level \(k\) or a nonabelian theta function. The space \(H^0(M_g,\mathcal{L}^k)\) can be identified (see the work of A. Beauville and Y. Laszlo) with the space of conformal block of level \(k\) of the SU(2) Wess-Zumino-Witten model for wich the author describes an abelianization procedure, carrying out the program of M.\ Atiyah and N.\ Hitchin. To be more precise, the author gives an explicit representation of a basis of \(H^0(M_g,\mathcal{L}^k)\) and its transformation formula in terms of classical Riemann theta functions of degree \(k+2\) with automorphic form coefficients, defined on the Prym variety associated with a two-fold branched covering surface. Using pants decomposition, the first section gives some topological properties of a family of 2-fold branched covering surfaces \(\widetilde{C}\) of \(C\) parametrized by the configuration space of \(4g-4\) unordered (mutually distinct) points. Now one considers the Prym varieties (more precisely the translated Prym varieties \(P'=\{ \widetilde{L}\in J^{2g-2}(\widetilde{C}) : \text{Nm}(\widetilde{L})=\omega_C\}\)) which can be equipped with natural coordinates coming from the pant decomposition. If we denote \(P\) the natural covering space of \(P'\), the author studies the morphism \(\pi:P\rightarrow M_g\) (which is a dominant map) and a pulled-back section of \(H^0(M_g,\mathcal{L}^k)\) can be expressed by Riemann theta functions of level \(2k\) on \(P\). A key point here is the description of the action induced by the fundamental group of the configuration space on the space of Riemann theta functions on \(P\). This leads to consider the notion of projectively invariant Riemann theta functions under this act…
reviewed by: Julien Keller
Mathematics Subject Classification ID: 14H60
Mathematics Subject Classification ID: 14D20
Mathematics Subject Classification ID: 14H40
Mathematics Subject Classification ID: 14D21
Mathematics Subject Classification ID: 53D50
Mathematics Subject Classification ID: 53C07
zbMATH DE Number: 5081752
zbMATH Keywords: moduli of semistable vector bundle
zbMATH Keywords: SU(2) WZW model
zbMATH Keywords: non abelian theta function
MaRDI profile type: Publication
full work available at URL: annals.2006.164.1
OpenAlex ID: W2006759418
DOI: 10.4007/ANNALS.2006.164.1
description: scientific article; zbMATH DE number 5081752